Notre Dame Journal of Formal Logic

Model Companions of $T_{\rm Aut}$ for Stable T

John T. Baldwin and Saharon Shelah

Abstract

We introduce the notion T does not omit obstructions. If a stable theory does not admit obstructions then it does not have the finite cover property (nfcp). For any theory T, form a new theory $T_{\rm Aut}$ by adding a new unary function symbol and axioms asserting it is an automorphism. The main result of the paper asserts the following: If T is a stable theory, T does not admit obstructions if and only if $T_{\rm Aut}$ has a model companion. The proof involves some interesting new consequences of the nfcp.

Article information

Source
Notre Dame J. Formal Logic Volume 42, Number 3 (2001), 129-142.

Dates
First available: 12 September 2003

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1063372196

Digital Object Identifier
doi:10.1305/ndjfl/1063372196

Mathematical Reviews number (MathSciNet)
MR2010177

Zentralblatt MATH identifier
1034.03040

Subjects
Primary: 03C45: Classification theory, stability and related concepts [See also 03C48]

Keywords
stability expansion by automorphism

Citation

Baldwin, John T.; Shelah, Saharon. Model Companions of $T_{\rm Aut}$ for Stable T . Notre Dame Journal of Formal Logic 42 (2001), no. 3, 129--142. doi:10.1305/ndjfl/1063372196. http://projecteuclid.org/euclid.ndjfl/1063372196.


Export citation

References

  • [1] Baldwin, J. T., Fundamentals of Stability Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1988.
  • [2] Chatzidakis, Z., and A. Pillay, "Generic structures and simple theories", Annals of Pure and Applied Logic, vol. 95 (1998), pp. 71--92.
  • [3] Chatzidakis, Z., and U. Hrushovski, "The model theory of difference fields", Transactions of AMS, vol. 351 (1999), pp. 2997--3071.
  • [4] Kikyo, H., "Model companions of theories with an automorphism", The Journal of Symbolic Logic, vol. 65 (2000), pp. 1215--22.
  • [5] Kikyo, H., and A. Pillay, "The definable multiplicity property and generic automorphisms", Annals of Pure and Applied Logic, vol. 106 (2000), pp. 263--73.
  • [6] Kikyo, H., and S. Shelah, "The strict order property and generic automorphisms", The Journal of Symbolic Logic, vol. 67 (2002), pp. 214--16.
  • [7] Lascar, D., "Les beaux automorphismes", Archive for Mathematical Logic, vol. 31 (1991), pp. 55--68.
  • [8] Pillay, A., "Notes on model companions of stable theories with an automorphism", http://www.math.uiuc.edu/People/pillay.html, 2001.
  • [9] Shelah, S., Classification Theory and the Number of Nonisomorphic Models, 2d edition, Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
  • [10] Shelah, S., "On model completion of $T_\rm aut$", preprint, 2003.
  • [11] Winkler, P., "Model completeness and Skolem expansions", pp. 408--64 in Model Theory and Algebra: A Memorial Tribute to Abraham Robinson, edited by D. H. Saracino and V. B. Weispfenning, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1975.